What's in a map?

For some reason or another, maps and mapping have always been a fascinating topic for me. As far back as I can remember, I've been astonished to see those miniaturized copies of the world (now other worlds) around me, on a modest-sized piece of paper. Just by looking at one you can figure out where you are probably located, and from there what direction to head to get somewhere else, toward a distant goal, most likely a goal depicted on the same sheet. (Of course these days we also have GPS, but one still needs good maps!)
 
Eventually you figure out all the symbols, the ones for Interstate driving maps, or topographic maps a geologist might need, or specialized maps designed to show the population density in every part of the world. IN a good Atlas you may also find maps drawn with strange "projections" (as they're called) that show all the land masses correctly related to each another, all small regions in correct shape (nearly no distortion), but some places are enlarged greatly, while others appear shrunken. A different kind may show all the World's areas in correct proportion to each other, but this time the shapes will be distorted depending on where you look. The biggest errors of either type usually occur away from the center -- the middle being pretty good on most maps you're likely to encounter.
 
By the mid-'70's I wanted to learn more of the details about how maps are designed and made. What exactly was "a projection"? There are books that will explain some of these things to you, at a very modest beginner's level (suitable even for children,) or at a professional level that will likely require some knowledge of math, to ferret out the details. I kinda fell in the middle, just where the best books were NOT. It took study of some tough chapters, but only with reference to the easier books, before I began to grasp a little of this fairly intricate field.
 
Of course the whole thing would be a lot easier if the Church elders and other fools of a distant time had been right about the Earth being flat. A flat Earth would map very easily onto a flat piece of papyrus or parchment. When we face up to our spherical planet (actually it's an ellipsoidal shape, but only slightly so), the best map ought be one printed on a sphere. Don't laugh, these DO exist your know -- they're called Globes (ho-ho!) But for most purposes we have to take a deep breath and do something about pushing the features of a round planet onto a flat sheet. That's the reason for "projecting" a curved reality onto a flat surface You really can't just chop out a piece of lower Broadway, flatten it out in an hydraulic press, and shrink it to size. Just measure the stuff you want to put on the map, and then find a way to depict this on a paper sheet with a minimum of distortion and alterations (except size -- consider a 1:1 full sized "map"... :-)
 
This is not the place to go into more than this superficial overview. There's a list of some good map books at the bottom of this page, for any of you who want to learn more, to try yourself, or just to appreciate this honorable, inventive art form. Today most drafting of map projections is done via computer. Final human handwork may also use a fancy machine with a graphics tablet, but in other cases traditional drafting tools can embellish and complete the raw plotter output.
 
When I became involved, I was lucky to have recently acquired a early nifty little home computer, that I used in computer music and synthesis. It was the Hewlett-Packard HP 9825A. I still have it, it still works (my microtuning table are often generated there to send to the synths), and there's comfort in knowing that it once helped me to learn a bit about mapping and math, and let me create some original maps that you'll see below.

A Difficult Mapping

In much the same way that my Eclipse Page started small and then expanded slowly over the past three years, let's begin with a few examples of some interesting (well, they are to me, anyway... ;-) maps and their projections. In all of them the actual land forms are from a database of the world's coastal outlines and country boundaries. At the time I began, 1976, there were no compact sources I could find for such things -- home computers had just begun to appear -- so I had to build my own.
 
That required hand-measuring many points on several good maps in Latitude (the North-South direction) and Longitude (the East-West one,) and typing these numbers into the computer. This became a chore very quickly, trying to "eyeball" the points that best represented the turns and undulations of the general coastal outline. I eventually invented a way to use the HP plotter, positioning a dummy pen at each likely spot, and then sending that to the computer. For that you have to also figure out a program to convert the X and Y plotter values into those necessary Lat and Long positions.
 
Then you write program to plot these values, with some grid lines and an outline shape for the Earth, or an outer boundary for a close-up map. That was the most fun, trying out various ways to "project" my little data base (about 2000 points -- the computer had 24Kilobytes of user RAM -- imagine that!) into some recognizable map, or eventually new ones. Let's look at the most difficult traditional one I came across. It's a general, tiltable (in any direction) and scalable variation on the attractive Mollweide Projection, which means it is an Oblique Mollweide.
 

 
Here's what it might look like, in an interesting orientation, one that neatly depicts the entire Earth on a single sheet. Distortion is acceptably low in the areas we most are interested in, the main land masses. It also has zero area distortion, while it shows the way the three major land groups are located in relationship to one another (please click the small map to obtain a large view, suitable for printing. You may have to scroll it a little on your screen.)
 
What makes this lovely, modest map so difficult to plot with computer? The Mollweide projection involves an equality that turns out to be transcendental, and unfortunately can't be solved explicitly (cartographers often construct a graph, and then read off values from the curve, a clumsy solution, I thought.) We'll spare the actual details here for now. But I can tell you that I found my solution via an extremely close approximation, one that had to be tested and tweaked by trial and error, using the HP computer and plotter together for a few days, to get it just right.
 
The final equation is an exponential function that calculates quickly. It cranks out values that plot accurate to within 1/10th the width of the pen tip at the worst case (small regions near the very top and bottom,) and considerably less everywhere else. That certainly seemed pretty acceptable, when I couldn't SEE or measure the errors on the final 11" x 17" plot paper sheets... Pragmatism strikes again! With this "almost exact" way to calculate the Y values, you then need only add some rotational terms to the program (for the oblique property), and then optimize all the code to be as fast as possible to calculate. The HP 9825 was no speed demon -- a 16-bit (count 'em) CPU chip going at but a couple of MHz. Today any typical machine could dance thousands of circles around it each cycle, and clever programming tricks for speed would no longer be so necessary (though it was part of the challenge back then, *sigh*...)

A Deceptive Map

This next example contains a sneaky trick or two. It's been made to look three-dimensional, and oriented here to give you an unsubtle view of the trap. If I held such a globe in front of you, untilted, and perhaps gave it a very slow complete turn in my hands, and asked you what you saw, what would you describe? That it's obviously a globe of the Earth, a pretty blue and beige one. Kinda shiny. And wot else...? (This top view was deliberately plotted to make what is going on here rather obvious...) It's funny that many people without a lot of geographical and map experience will miss spotting at once that this is a double Earth -- every location appears twice!
 
The motive for working out a way to construct this mapping came from one of Martin Gardner's excellent columns in Scientific American magazine, which he wrote for several decades. The 1976 article, about several provocative new mapping ideas, was included in a collection of these Mathematical Games columns: "Time Travel and Other Mathematical Bewilderments" (1988), chpt. 15. Gardner had visited a mathematician at Bell Labs in the 70's, Edgar N. Gilbert. It was Gilbert who first designed and built such a globe as described above. Hooked by curiosity, I had to dope out how it was done (the article didn't go into any of the math), and try to deduce what must have gone into it.
 

 
This entailed several wasted "insights" over a couple of weeks before I stumbled upon a decent "eureka!", and was able to crack the puzzle. As I had been plotting maps for a year or two, it was possible to try out the idea, printing up several different views (like above) of such a globe. There is no real globe involved of course. The program takes my earth data base, and "plots" the points onto a virtual Mercator-mapped Cylinder, then splits the cylinder vertically, and splices in a clone, yielding a double circumference cylinder. Then it halves all the dimensions to get back the original circumference (Mercator projections are theoretically infinitely tall, so half of infinity is still infinity...), and then inverses-plots this back to a blank virtual globe. Finally the points on the new double earth globe are mapped off to a large paper sheet via some Oblique Orthographic plotting equations (actually all the steps are done at one time in a program optimized for speed, though it still takes a while on the old HP.)
 
The image you see above was obtained from two scans of the original oversized plot, stitched together, cleaned up, and then turned into a dimensional globe using Photoshop's Airbrush Tool, and then a lot of gentle massaging with the Burn and Dodge Tools, manually with my trusty Wacom tablet. I replaced the original plotted descriptions at the bottom with new ones, made with some of Photoshop's neat 3-D text effects -- who could resist?

Hole-Digger's Map

Now let's look at a much more practical map. This one is designed to show you, very quickly and easily, something that friends of mine had spoken about since I was growing up. Perhaps we are all influenced by the old saw that jokes about "digging a whole all the way to China". Have you ever thought about what that really means? Not only about the Earth's core being molten iron and such. About where you actually might come out if you could perfect your digging prowess to an astonishing degree?
 

 
The map above is projected with a baby-simple world map that I "invented", just to say I had created one of those rectangular all-Earth maps with the best compromises of shape and area. I call it "Carlos Omni", and it works quite successfully. The outer shape is flat on top and bottom, but curved with a cosine function on both sides. In this version the Earth database got plotted once in black ink. As a second step I flipped the whole thing from top to bottom (note it's not rotated, West is still to the left), so everything's upside down, but it begins with a 180 degrees shift in Longitude compared to the black version.
 
That forces both the horizontal and vertical points to differ by 180 degrees, which is the definition of the locations exactly opposite one another on a globe. Note that the full size version is fairly large and high res, suitable for printing. You may have to scroll it a bit on your screen. The tiny instructions printed on the bottom of the map suggest:
 

Find the point directly opposite you on the other side of the Earth. For a location on the black outline map, that spot on the inverted red map (or vice-versa) will be your Local Nadir, or Antipode.

Examples: Spain is opposite New Zealand's North Island. The center of Brazil is opposite The Philippines. Botswana is opposite the Hawaiian Islands.

 
Even if you own a good shovel (and could dig perfectly straight down -- dig at a random angle and all bets are off), you probably won't have a great many uses for a Local Nadir or Antipodes map (I called it the first name originally, being technically precise, and then realized the second name describes it in more common terminology as well), but it still may be surprising to most of you, and might even help in settling a few bets...!

(P.S. Now you can see that if your mom or dad or friends HAD been right about digging straight down "all the way to China," you must have grown up in Chile or Argentina... Q.E.D.)

A Map as Cover-Art

Here's a subtle map projection of my own that for a while I entertained using as the background image on a new album of music I was then composing. The album underwent a fairly lengthy path before being completed, and it accrued a few temporary titles. Once Upon An Earth was one of the first, as I recall. My generous pen-pal Arthur C. Clarke's later gave me permission to use one of his book titles from one of my favorite novels of his: "Songs of Distant Earth". What else to call an album inspired by the music of our world (this well before "world music" was even a category -- I seem to be always cursed as ahead of my time, dammitt..)
 
Can you guess which album of mine it turned into? It will be being remastered and released on ESD near the end of this year, 1999. You may have heard of it, but been unable to find a copy. A guess? The title: "Beauty in the Beast". Of course the final cover for BitB came out very differently from what an original map design would have looked like in a polished, final version. Since the music had taken on other different directions in subtle ways, too, as I composed it, that was only fair enough.

 
The image you see is one I got long before having such neat accessories as Photoshop and a flatbed scanner. So the top printing came from an early Apple Imagewriter printer, using a ribbon so the quality is poor. It was designed in my first Mac Plus, using MacPaint (wot else?) The map was a reduced size photocopy of the original large plot that came from my HP 9872A four-pen flatbed plotter, driven with the HP 9825A. I quickly filled in the land forms with a soft lead pencil, which still shows up. The superimposition then was made in a copy machine. For this view I just scanned in the two original sheets and superimposed them in Photoshop, and retouched it a little.

The Map Above

Let's take a closer look at the map used in the mockup album cover above. Since I invented it, I get to call it the: "Carlos Multi-Conformal Projection". This example is centered in the usual spot for a map of the whole Earth, at zero Latitude and zero Longitude. Like other maps of this kind, the Van der Grinten being one of the more popular versions, the whole surface is placed in a circular outline. But where most other historic maps of this kind use projections that arose in an arithmetic attempt to put everything more or less in its place, this one has an advantage or two over them.
 
Unlike the others, which are neither conformal, nor the somewhat easier equal area (the first means shapes are preserved by allowing the size-scale to be inconstant, the second means areas are preserved by allowing the shapes to be inconstant), this one is conformal. Not many other examples of this kind exist. Also, notice that the outer margin is NOT a circle, but is wider than the height. The amount of this "overscan" is controlled in the initial setup to plot the map, and so is the scale, which allows either the whole thing to fit on one sheet, or only an enlarged portion, if that be desired.

 
You can look at this higher resolution image by clicking the smaller one above. You'll note that any small region indeed does look like an overhead flat mapping of that location, normal to the surface. So Alaska looks true to its form, as does each other single small country. Only land plotted at the poles gets "torn" and spread open, but the margin again remains conformally matched to the places depicted. Antarctica is the only "torn" example seen in this view. Note that sizes increase outwards, so places near the center are smaller than the average scale, while those out near the edge are enlarged, as the famous Mercator Projection maps we grew up with. But the scale is more constant here, thanks to the way the outer portions "wrap around" inside the round outer shape, not expanding off into infinity as Mercator style maps do at top and bottom.
 
Of course I'm only a self-taught amateur, and was not in contact with professional map makers when I invented the above projection. I thought it was quite unique. It's not, of course -- there are a very few related projections that have been around for years, but that I was unaware of when I came up with the idea. Lagrange and Eisenlohr Projections both come very close, although the details are not exact, and the equations used are different. Well, sure, I self-taught myself into an elaborate scheme of doing what looked like a neat idea -- had I seen either of these two projections first, I probably would not have bothered at all.
 
We'll get some other examples from the stack of HP Plotter sheets and present them to you here another time. Let me see if I can also locate some of the final optimized mapping equations and computer code, and put these on a co-page, for reference. If your curiosity is already piqued, you may want to look up several texts that were quite helpful to me, and might be for you. You can check in your local library, otherwise they may have to be specially ordered for you, as I did.
 
The top book is easily available, and makes a wonderful layman's introduction to the whole topic and related arts. The second book on the list is a winner if you need a quick background, want to look at lots of plots and learn trivia about them and the equations used for each. It's also an inexpensive bargain available through the Government Printing Offices. The remaining three are college type textbooks, and will be of more help if you have some mathematical understanding. They all had a lot of good details in them, but I found the Mailing text to be the most useful. You'll have to dope out the ways to get the math concepts into practical forms you can use in a small computer, and that was a great deal of the fun, and why I stuck with it, as I learned a lot more by doing or trying to do than any textbook can tutor just by reading alone.

Title	Author(s)	Date	Publisher
Mapping	David Greenhood	1964	Phoenix #PS5521
An Album of Map Projections	Snyder and Voxland	1989/94	USGS Pro Paper 1453
Coordinate Sys & Map Projections	D.H Maling	1973	Philip & Sons, London
Intro to Study of Map Projections	J. A. Steers	1970	University of London Press
Map Projections	Richardus & Adler	1972	North-Holland Publishing